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Instability in the Gel’fand inverse problem at high energies
Mikhail Isaev
To cite this version:
Mikhail Isaev. Instability in the Gel’fand inverse problem at high energies. Applicable Analysis,
Taylor & Francis, 2012, pp.DOI:10.1080/00036811.2012.731501. �10.1080/00036811.2012.73150�. �hal-
00706957�
Instability in the Gel’fand inverse problem at high energies
M.I. Isaev
Abstract
We give an instability estimate for the Gel’fand inverse boundary value problem at high energies. Our instability estimate shows an optimality of several important preceeding stability results on inverse problems of such a type.
1 Introduction
In this paper we continue studies on the Gel’fand inverse boundary value prob- lem for the Schr¨ odinger equation
− ∆ψ + v(x)ψ = Eψ, x ∈ D, (1.1)
where
D is an open bounded domain in R
d, d ≥ 2,
with ∂D ∈ C
2, (1.2)
v ∈ L
∞(D). (1.3)
As boundary data we consider the map ˆ Φ = ˆ Φ(E) such that Φ(E)(ψ ˆ |
∂D) = ∂ψ
∂ν |
∂D(1.4)
for all sufficiently regular solutions ψ of (1.1) in ¯ D = D ∪ ∂D, where ν is the outward normal to ∂D. Here we assume also that
E is not a Dirichlet eigenvalue for operator − ∆ + v in D. (1.5) The map ˆ Φ = ˆ Φ(E) is known as the Dirichlet-to-Neumann map.
We consider the following inverse boundary value problem for equation (1.1):
Problem 1.1. Given ˆ Φ for some fixed E, find v.
This problem is known as the Gel’fand inverse boundary value problem for the Schr¨odinger equation at fixed energy (see [7], [19]). At zero energy this problem can be considered also as a generalization of the Calderon problem of the electrical impedance tomography (see [5], [19]). Problem 1.1 can be also considered as an example of ill-posed problem: see [14], [3] for an introduction to this theory.
There is a wide literature on the Gel’fand inverse problem at fixed energy.
In a similar way with many other inverse problems, Problem 1.1 includes, in
particular, the following questions: (a) uniqueness, (b) reconstruction, (c) sta- bility.
Global uniqueness results and global reconstruction methods for Problem 1.1 were obtained for the first time in [19] in dimension d ≥ 3 and in [4] in dimension d = 2.
Global logarithmic stability estimates for Problem 1.1 were obtained for the first time in [1] in dimension d ≥ 3 and in [25] in dimension d = 2. A principal improvement of the result of [1] was obtained recently in [24] (for the zero energy case): stability of [24] optimally increases with increasing regularity of v.
Note that for the Calderon problem (of the electrical impedance tomography) in its initial formulation the global uniqueness was firstly proved in [30] for d ≥ 3 and in [17] for d = 2. Global logarithmic stability estimates for this problem were obtained for the first time in [1] for d ≥ 3 and [15] for d = 2. Principal increasing of global stability of [1], [15] for the regular coefficient case was found in [24] for d ≥ 3 and [28] for d = 2. In addition, for the case of piecewise real analytic conductivity the first uniqueness results for the Calderon problem in dimension d ≥ 2 were given in [13]. Lipschitz stability estimate for the case of piecewise constant conductivity was obtained in [2] (see [27] for additional studies in this direction).
The optimality of the logarithmic stability results of [1], [15] with their princi- pal effectivizations of [24], [28] (up to the value of the exponent) follows from [16].
An extention of the instability estimates of [16] to the case of the non-zero en- ergy as well as to the case of Dirichlet-to-Neumann map given on the energy intervals was obtained in [8].
On the other hand, it was found in [20], [21] (see also [23], [26]) that for inverse problems for the Schr¨ odinger equation at fixed energy E in dimension d ≥ 2 (like Problem 1.1) there is a H¨ older stability modulo an error term rapidly decaying as E → + ∞ (at least for the regular coefficient case). In addition, for Problem 1.1 for d = 3, global energy dependent stability estimates changing from logarithmic type to H¨ older type for high energies were obtained in [12], [11]. However, there is no efficient stability increasing with respect to increasing coefficient regularity in the results of [12]. An additional study, motivated by [12], [24], was given in [18].
The following stability estimate for Problem 1.1 was recently proved in [11]:
Theorem 1.1 (of [11]). Let D satisfy (1.2), where d ≥ 3. Let v
j∈ W
m,1(D), m > d, supp v
j⊂ D and || v
j||
Wm,1(D)≤ N for some N > 0, j = 1, 2, (where W
m,pdenotes the Sobolev space of m-times smooth functions in L
p). Let v
1, v
2satisfy (1.5) for some fixed E ≥ 0. Let Φ ˆ
1(E) and Φ ˆ
2(E) denote the DtN maps for v
1and v
2, respectively. Let s
1= (m − d)/d. Then, for any τ ∈ (0, 1) and any α, β ∈ [0, s
1], α + β = s
1,
|| v
2− v
1||
L∞(D)≤ A(1 + √
E)δ
τ+ B(1 + √
E)
−αln 3 + δ
−1−β, (1.6) where δ = || Φ ˆ
2(E) − Φ ˆ
1(E) ||
L∞(∂D)→L∞(∂D)and constants A, B > 0 depend only on N , D, m, τ.
In particular cases, H¨ older-logarithmic stability estimate (1.6) becomes co-
herent (although less strong) with respect to results of [21], [23], [24]. In this
connection we refer to [11] for more detailed infromation. Concerning two-
dimensional analogs of results of Theorem 1.1, see [20], [26], [28], [29].
In a similar way with results of [9], [10], estimate (1.6) can be extended to the case when we do not assume that condition (1.5) is fulfiled and consider an appropriate impedance boundary map (or Robin-to-Robin map) instead of the Dirichlet-to-Neumann map.
In the present work we prove optimality of estimate (1.6) (up to the values of the exponents α, β) in dimension d ≥ 2. Our related instability results for Problem 1.1 are presented in Section 2, see Theorem 2.1 and Proposition 2.1.
Their proofs are given in Section 4 and are based on properties of solutions of the Schr¨ odinger equation in the unit ball given in Section 3.
2 Main results
In what follows we fix D = B
d(0, 1), where
B
d(x
0, ρ) = { x ∈ R
d: || x − x
0||
Ed< ρ } , x
0∈ R
d, ρ > 0. (2.1) Let || F || denote the norm of an operator
F : L
∞(∂D) → L
∞(∂D). (2.2) We recall that if v
1, v
2are potentials satisfying (1.3), (1.5) for some fixed E, then Φ ˆ
2(E) − Φ ˆ
1(E) is a compact operator in L
∞(∂D), (2.3) where ˆ Φ
1, ˆ Φ
2are the DtN maps for v
1, v
2, respectively, see [19], [22].
Our main result is the following theorem:
Theorem 2.1. Let D = B
d(0, 1), where d ≥ 2. Then for any fixed constants A, B, κ, τ, ε > 0, m > d and s
2> m there are some energy level E > 0 and some potential v ∈ C
m(D) such that condition (1.5) holds for potentials v and v
0≡ 0, simultaneously, supp v ⊂ D, k v k
L∞(D)≤ ε, k v k
Cm(D)≤ C
1, where C
1= C
1(d, m) > 0, but
|| v − v
0||
L∞(D)> A(1 + √
E)
κδ
τ+ B (1 + √
E)
2(s−s2)ln 3 + δ
−1−s(2.4) for any s ∈ [0, s
2], where Φ, ˆ Φ ˆ
0are the DtN map for v and v
0, respectively, and δ = || Φ(E) ˆ − Φ ˆ
0(E) || is defined according to (2.2).
Theorem 2.1 shows, in particular, the optimality (at least for potentials in the neighborhood of zero) of estimate (1.6) (up to the values of the exponents α, β). As a corollary of Theorem 2.1, one can obtain an optimality of the stability results of [20], [21], [23], [26].
In the present work Theorem 2.1 is proved by explicit instability example with complex potentials. Examples of this type were considered for the first time in [16] for showing the exponential instability in Problem 1.1 in the zero energy case. An extention to the case of the non-zero energy as well as to the case of Dirichlet-to-Neumann map given on the energy intervals was obtained in [8].
Let us consider the cylindrical variables:
(r
1, θ, x
′) ∈ R
+× R /2π Z × R
d−2, r
1cos θ = x
1, r
1sin θ = x
2, x
′= (x
3, . . . , x
d).
(2.5)
Take φ ∈ C
∞( R
2) with support in B
2(0, 1/3) ∩{ x
1> 1/4 } and with k φ k
L∞= 1.
For integers m, n > 0, define the complex potential
v
nm= n
−me
inθφ(r
1, | x
′| ). (2.6) We recall that
k v
nmk
L∞= n
−m, k v
nmk
Cm≤ C
1, (2.7) where C
1= C
1(d, m) > 0. Note that C
1is the same as in Theorem 2.1.
Estimates (2.7) were given in [16] (see Theorem 2 of [16]).
To prove Theorem 2.1 we use, in partucular, the following proposition:
Proposition 2.1. Let D = B
d(0, 1), where d ≥ 2. Let condition (1.5) hold with v ≡ v
nm(of (2.6)) and v ≡ v
0≡ 0 for some E > 0 and some integers m > 0, n > 20(1 + √
E)
2. Then, for any σ > 0,
k Φ ˆ
nm(E) − Φ ˆ
0(E) k
H−σ(Sd−1)→Hσ(Sd−1)≤ C
2(1 + Q + EQ)2
−n/4, (2.8) where Φ ˆ
nm, Φ ˆ
0are the DtN map for v
nmand v
0, respectively, C
2= C
2(d, σ) > 0, Q = k ( − ∆ + v
0− E)
−1k
L2(D)→L2(D)+ k ( − ∆ + v
nm− E)
−1k
L2(D)→L2(D), (2.9) where ( − ∆ + v
0− E)
−1, ( − ∆ + v
nm− E)
−1are considered with the Dirichlet boundary condition in D and H
±σ= W
±σ,2denote the standart Sobolev spaces.
Analogs of estimate (2.8) (but without dependence of the energy) were given in Theorem 2 of [16] for the zero energy case and in Theorem 2.4 of [8] for the case of the non-zero energy and the case of the energy intervals.
We obtain Theorem 2.1, combining known results on the spectrum of the Laplace operator in the unit ball (see formula (4.9) below), Proposition 2.1, estimates (2.7) and the fact that
k F k
L∞(Sd−1)→L∞(Sd−1)≤ c(d, σ) k F k
H−σ(Sd−1)→Hσ(Sd−1)(2.10) for sufficiently large σ. The detailed proof of Theorem 2.1 and the proof of Proposition 2.1 are given in Section 4. These proofs use, in particular, results, presented in Section 3.
Remark 2.1. In a similar way with [16], [8], using a ball packing and covering by ball arguments (see also [6]), the instability result of Theorem 2.1 can be extended to the case when only real-valued potentials are considered and in the neighborhood of any potential (not only v
0≡ 0).
3 Some properties of solutions of the Schr¨ odinger equation in the unit ball
In this section we continue assume that D = B
d(0, 1), where d ≥ 2. We fix an orthonormal basis in L
2( S
d−1) = L
2(∂D)
{ f
jp: j ≥ 0, 1 ≤ p ≤ p
j} ,
f
jpis a spherical harmonic of degree j, (3.1)
where p
jis the dimension of the space of spherical harmonics of order j, p
j=
j + d − 1 d − 1
−
j + d − 3 d − 1
, (3.2)
where
n k
= n(n − 1) · · · (n − k + 1)
k! for n ≥ 0 (3.3)
and n
k
= 0 for n < 0. (3.4)
The precise choice of f
jpis irrelevant for our purposes. Besides orthonormality, we only need f
jpto be the restriction of a homogeneous harmonic polynomial of degree j to the sphere and so | x |
jf
jp(x/ | x | ) is harmonic. We use also the polar coordinates (r, ω) ∈ R
+× S
d−1, with x = rω ∈ R
d.
Lemma 3.1. Let D = B
d(0, 1), where d ≥ 2. Let potential v satisfy (1.3) and (1.5) for some fixed E. Let || v ||
L∞(D)≤ N, for some N > 0. Then for any solution ψ ∈ C(D ∪ ∂D) of equation (1.1) the following inequality holds:
k ψ k
L2(D)≤
1 + (N + | E | ) k ( − ∆ + v − E)
−1k
L2(D)→L2(D)k f k
L2(∂D), (3.5) where f = ψ |
∂D, ( − ∆ + v − E)
−1is considered with the Dirichlet boundary condition in D.
Proof of Lemma 3.1. We expand the function f in the basis { f
jp} : f = X
j,p
c
jpf
jp. (3.6)
We have that
k f k
2L2(∂D)= X
j,p
| c
jp|
2. (3.7)
Let
ψ
0(x) = X
j,p
c
jpr
jf
jp(ω). (3.8)
Note that
k ψ
0k
2L2(D)= X
j,p
| c
jp|
2k r
jf
jp(ω) k
2L2(D)=
= X
j,p
| c
jp|
2Z
10
r
2j+d−1dr ≤ X
j,p
| c
jp|
2(3.9)
Using (1.1) and the fact that ψ
0is harmonic, we get that
( − ∆ + v − E)(ψ − ψ
0) = (E − v)ψ
0. (3.10) Since ψ |
∂D= ψ
0|
∂D= f , using (3.10), we find that
k ψ − ψ
0k
L2(∂D)≤ (N + | E | ) k ( − ∆ + v − E)
−1k
L2(D)→L2(D)k ψ
0k
L2(D). (3.11)
Combining (3.7), (3.9), (3.11), we obtain (3.5).
Let < · , · > denote the scalar product in the Hilbert space L
2(∂D):
< f, g >=
Z
∂D
f (x)¯ g(x)dx, (3.12)
where f, g ∈ L
2(∂D).
Lemma 3.2. Let D = B
d(0, 1), where d ≥ 2. Let potentials v
1, v
2satisfy (1.3) and (1.5) for some fixed E. Let v
1, v
2be supported in B
d(0, 1/3) and
|| v
i||
L∞(D)≤ N , i = 1, 2, for some N > 0. Then for any j
1, j
2∈ N ∪ { 0 } , 1 ≤ p
1≤ p
j1, 1 ≤ p
2≤ p
j2and j
max= max { j
1, j
2} ≥ 10(1 + p
| E | )
2the following inequality holds:
D f
j1p1,
Φ ˆ
1(E) − Φ ˆ
2(E) f
j2p2E
≤ C(d)
1 + (N + | E | )Q
2
−jmax, (3.13) where
Q = k ( − ∆ + v
1− E)
−1k
L2(D)→L2(D)+ k ( − ∆ + v
2− E)
−1k
L2(D)→L2(D), (3.14) Φ ˆ
1, Φ ˆ
2are the DtN map for v
1and v
2, respectively, and ( − ∆ + v
1− E)
−1, ( − ∆ + v
2− E)
−1are considered with the Dirichlet boundary condition in D.
Analogs of estimate (3.13) (but without dependence of the energy) were given in Lemma 1 of [16] for the zero energy case and in Lemma 3.4 of [8] for the case of the non-zero energy and the case of the energy intervals.
We prove Lemma 3.2 for E 6 = 0 in Section 5, using expression of solutions of equation − ∆ψ = Eψ in B
d(0, 1) \ B
d(0, 1/3) in terms of the Bessel functions J
αand Y
αwith integer or half-integer order α.
4 Proofs of Proposition 2.1 and Theorem 2.1
We continue to assume that D = B
d(0, 1), where d ≥ 2 and to use the orthonor- mal basis { f
jp: j ∈ N ∪ { 0 } , 1 ≤ p ≤ p
j} in L
2( S
d−1) = L
2(∂D). The Sobolev spaces H
σ( S
d−1) can be defined by
X
j,p
c
jpf
jp:
X
j,p
c
jpf
jpHσ
< + ∞
,
X
j,p
c
jpf
jp2
Hσ
= X
j,p
(1 + j)
2σ| c
jp|
2,
(4.1)
see, for example, [16].
Consider an operator A : H
−σ( S
d−1) → H
σ( S
d−1). We denote its matrix elements in the basis { f
jp} by
a
j1p1j2p2=< f
j1p1, Af
j2p2> . (4.2)
We identify in the sequel an operator A with its matrix { a
j1p1j2p2} . In this
section we always assume that j
1, j
2∈ N ∪ { 0 } , 1 ≤ p
1≤ p
j1, 1 ≤ p
2≤ p
j2.
We recall that (see formula (12) of [16]) k A k
H−σ(Sd−1)→Hσ(Sd−1)≤ 4 sup
j1,p1,j2,p2
(1 + max { j
1, j
2} )
2σ+d| a
j1p1j2p2| . (4.3) Proof of Proposition 2.1. In a similar way with the proof of Theorem 2 of [16]
we obtain that
< f
j1p1,
Φ ˆ
mn(E) − Φ ˆ
0(E)
f
j2p2>= 0 (4.4) for j
max= max { j
1, j
2} ≤
n−12
(the only difference is that instead of the operator − ∆ we consider the operator − ∆ − E), where [ · ] denotes the integer part of a number. Note that
n − 1 2
+ 1 ≥ n/2 > 10(1 + √
E)
2, k v
nmk
L∞(D)≤ 1. (4.5) Combining (4.3), (4.4), (4.5) and Lemma 3.2, we get that
k Φ ˆ
mn(E) − Φ ˆ
0(E) k
H−σ(Sd−1)→Hσ(Sd−1
)
≤
≤ 4C(d)
1 + (1 + E)Q sup
jmax≥n/2
(1 + j
max)
2σ+d2
−jmax≤
≤ C
2(d, σ)(1 + Q + EQ)2
−n/4,
(4.6)
where
Q = k ( − ∆ + v
0− E)
−1k
L2(D)→L2(D)+ k ( − ∆ + v
nm− E)
−1k
L2(D)→L2(D). (4.7)
Let N (ρ) denote the counting function of the Laplace operator in D N(ρ) = |{ λ < ρ
2: λ is a Dirichlet eigenvalue of − ∆ in D }| , (4.8) where | · | is the cardinality of the corresponding set. We recall that according to the Weyl formula (of [31]):
N(ρ) ≤ c
1(d)ρ
d. (4.9)
Lemma 4.1. Let D = B
d(0, 1), where d ≥ 1. Then for any ρ > 1 there is some E = E(ρ) ∈ (ρ
2, 2ρ
2) such that the interval
E(ρ) − c
2ρ
2−d, E(ρ) + c
2ρ
2−d(4.10) does not contain Dirichlet eigenvalues of − ∆ in D, where c
2= c
2(d) > 0.
Proof of Lemma 4.1. We put c
2= 2
d−1/(c
1(d) + 1). Then we can select k disjoint intervals of the length 2c
2ρ
2−din the interval (ρ
2, 2ρ
2), where
k = ρ
22c
2ρ
2−d= [(c
1(d) + 1)ρ
d] > N (ρ). (4.11) Thus, we have that at least one of these intervals does not contain Dirichlet
eigenvalues of − ∆ in D = B
d(0, 1).
Proof of Theorem 2.1. Let E = E(ρ) be the number of Lemma 4.1 for some ρ > 1. Using (4.10), we find that the distance from E to the Dirichlet spectrum of the operator − ∆ in D is not less than c
2ρ
2−d. Using also that E ∈ (ρ
2, 2ρ
2), we get that
k ( − ∆ − E)
−1k
L2(D)→L2(D)≤ 1
c
2ρ
2−d≤ E
(d−2)/2/c
2, (4.12) where ( − ∆ − E)
−1is considered with the Dirichlet boundary condition in D.
Let
n = [20(1 + √
E)
2] + 1. (4.13)
Using (2.7) and (4.10), we find that the distance from E to the Dirichlet spec- trum of the operator − ∆ + v
nmin D is not less than c
2ρ
2−d− n
−m, where v
nmis defined according to (2.6). Since m > d and E ∈ (ρ
2, 2ρ
2), using (4.13), we get that
k ( − ∆ + v
nm− E)
−1k
L2(D)→L2(D)≤ c
3E
(d−2)/2, E = E(ρ), ρ ≥ ρ
1(d, m) > 1, c
3= c
3(d, m) > 0,
(4.14) where ( − ∆ + v
nm− E)
−1is considered with the Dirichlet boundary condition in D.
Combining Proposition 2.1 and estimates (2.10), (4.12), (4.14), we find that δ = k Φ ˆ
nm(E) − Φ ˆ
0(E) k
L∞(Sd−1)→L∞(Sd−1
)
≤ c
4E
d/22
−n/4, E = E(ρ), ρ ≥ ρ
1(d, m) > 1,
n = [20(1 + √
E)
2] + 1 c
4= c
4(d, m) > 0.
(4.15)
Since s
2> m, taking ρ big enough and using (4.15), we obtain the following inequalities:
n
−m< ε, (4.16)
A(1 + √
E)
κδ
τ< 1
2 n
−m, (4.17)
B(1 + √
E)
2(s−s2)ln 3 + δ
−1−s< 1 2 n
−m, 0 ≤ s ≤ s
2,
(4.18) where
E = E(ρ), n = [20(1 + √
E)
2] + 1. (4.19)
Combining (2.6), (2.7), (4.16) - (4.19), we get that A(1 + √
E)
κδ
τ+ B(1 + √
E)
2(s−s2)ln 3 + δ
−1−s<
< 1
2 n
−m+ 1
2 n
−m= k v
nm− v
0k
L∞(D)k v
nmk
L∞(D)= n
−m< ε, k v
nmk
Cm(D)< C
1, supp v
nm⊂ D.
(4.20)
5 Proof of Lemma 3.2
To prove Lemma 3.2 we need some preliminaries. Consider the problem of finding solutions of the form ψ(r, ω) = R(r)f
jp(ω) of equation (1.1) with v ≡ 0 and D = B
d(0, 1), where d ≥ 2. We recall that:
∆ = ∂
2(∂r)
2+ (d − 1)r
−1∂
∂r + r
−2∆
Sd−1, (5.1) where ∆
Sd−1is Laplace-Beltrami operator on S
d−1,
∆
Sd−1f
jp= − j(j + d − 2)f
jp. (5.2) Then we obtain the following equation for R(r):
− R
′′− d − 1
r R
′+ j(j + d − 2)
r
2R = ER. (5.3)
Taking R(r) = r
−d−22R(r), we get ˜ r
2R ˜
′′+ r R ˜
′+ Er
2−
j + d − 2 2
2!
R ˜ = 0. (5.4)
This equation is known as the Bessel equation. For E = k
26 = 0 it has two linearly independent solutions J
j+d−22
(kr) and Y
j+d−22
(kr), where J
α(z) =
∞
X
m=0
( − 1)
m(z/2)
2m+αΓ(m + 1)Γ(m + α + 1) , (5.5) Y
α(z) = J
α(z) cos πα − J
−α(z)
sin πα for α / ∈ Z , (5.6)
and
Y
α(z) = lim
α′→α
Y
α′(z) for α ∈ Z . (5.7) We recall also that the system of functions
{ ψ
jp(r, ω) = R
j(k, r)f
jp(ω) : j ∈ N ∪ { 0 } , 1 ≤ p ≤ p
j} , is complete orthogonal system (in the sense of L
2) in the space of solutions of equation (1.1) in D
′= B(0, 1) \ B(0, 1/3)
with v ≡ 0, E = k
2and boundary condition ψ |
r=1= 0,
(5.8)
where
R
j(k, r) = r
−d−22Y
j+d−22
(kr)J
j+d−22
(k) − J
j+d−22
(kr)Y
j+d−22
(k)
. (5.9) For the proof of (5.8) see, for example, [8].
Lemma 5.1. For any ρ > 0, integers d ≥ 2, n ≥ 10(ρ + 1)
2and z ∈ C , | z | ≤ ρ, the following inequalities hold:
1 2
( | z | /2)
αΓ(α + 1) ≤ | J
α(z) | ≤ 3 2
( | z | /2)
αΓ(α + 1) , (5.10)
| J
α′(z) | ≤ 3 ( | z | /2)
α−1Γ(α) , (5.11)
1
2π ( | z | /2)
−αΓ(α) ≤ | Y
α(z) | ≤ 3
2π ( | z | /2)
−αΓ(α) (5.12)
| Y
α′(z) | ≤ 3
π ( | z | /2)
−α−1Γ(α + 1) (5.13) where
′denotes derivation with respect to z, α = n+
d−22and Γ(x) is the Gamma function.
In fact, the proof of Lemma 5.1 is given in [8] (see Lemma 3.3 of [8]). It was shown in [8] that inequalities (5.10) - (5.13) hold for any n > n
0, where n
0is such that
n
0> 3, exp
ρ
2/4 n
0+ 1
− 1 ≤ 1/2, 3π max 1, (ρ/2)
2n0+1Γ(n
0) + ρ
22n
0− ρ
2+ (ρ/2)
2n0e
ρ2/4Γ(n
0) ≤ 1/2,
(5.14)
(see formula (6.18) of [8]). The only thing to check is that n
0= [10(ρ + 1)
2] − 1 satisfy (5.14), where [ · ] denotes the integer part of a number, The first two inequalities are obvious. The third follows from the estimate
Γ(n
0) = (n
0− 1)! ≥
n
0− 1 e
n0−1. (5.15)
The final part of the proof of Lemma 3.2 consists of the following: first, we consider the case when E = k
26 = 0 and
j
1= max { j
1, j
2} ≥ 10(1 + | k | )
2. (5.16) Let ψ
1, ψ
2denote the solutions of equation (1.1) with boundary condition ψ |
∂D= f
j2p2and potentials v
1and v
2, respectively. Using Lemma 3.1 for v
1and v
2, we get that
k ψ
1− ψ
2k
L2(D)≤ 2
1 + (N + | E | )Q
, (5.17)
where
Q = k ( − ∆ + v
1− E)
−1k
L2(D)→L2(D)+ k ( − ∆ + v
2− E)
−1k
L2(D)→L2(D), (5.18) Note that ψ
1− ψ
2is the solution of equation (1.1) in D
′= B(0, 1) \ B(0, 1/3) with potential v ≡ 0 and boundary condition ψ |
r=1= 0. According to (5.8), we have that
ψ
1− ψ
2= X
j,p
c
jpψ
jpin D
′(5.19)
for some c
jp, where
ψ
jp(r, ω) = R
j(k, r)f
jp(ω). (5.20)
Since R
j(k, 1) = 0, we find that
∂R
j(k, r)
∂r
r=1= ∂
r
d−22R
j(k, r)
∂r
r=1
. (5.21)
For j ≥ 10(1 + | k | )
2, using Lemma 5.1, we have that
∂Ri(k,r)
∂r
r=1
Y
α(k)J
α(k)
= | k |
Y
α′(k)
Y
α(k) − J
α′(k) J
α(k) ≤
≤ 6 | k |
( | k | /2)
−α−1Γ(α + 1)
( | k | /2)
−αΓ(α) + ( | k | /2)
α−1Γ(α + 1) ( | k | /2)
αΓ(α)
= 6α,
(5.22)
|| r
−d−22Y
α(kr) ||
L2({1/3<|x|<2/5})| Y
α(k) |
2≥
≥ Z
2/51/3
1 3
( | k | r/2)
−αΓ(α) ( | k | /2)
−αΓ(α)
2r dr ≥ 2
5 − 1 3
1 3
1 3 (5/2)
α 2,
(5.23)
|| r
−d−22J
α(kr) ||
L2({1/3<|x|<2/5})| J
α(k) |
!
2≤
≤ Z
2/51/3
3 ( | k | r/2)
αΓ(α) ( | k | /2)
αΓ(α)
2r dr ≤ 2
5 − 1 3
1
3 (3(2/5)
α)
2,
(5.24)
where α = j +
d−22. Note that if j ≥ 10(1 + | k | )
2then j +
d−22> 3. Combining (5.23) and (5.24), we get that
|| ψ
jp||
L2({1/3<|x|<2/5})| Y
α(k)J
α(k) | ≥
≥ 2
5 − 1 3
1 3
1/21
3 (5/2)
α− 3(2/5)
α> 6
1000 (5/2)
α.
(5.25)
Combining (5.22) and (5.25), we find that
∂R
j(k, r)
∂r
r=1≤ 1000α(5/2)
−α|| ψ
jp(E) ||
L2({1/3<|x|<1}). (5.26) Proceeding from (5.19) and using the Cauchy-Schwarz inequality, we get that
| c
jp| =
D ψ
jp, ψ
1− ψ
2E
L2({1/3<|x|<1})
